Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,26,29,31)(27,30,33,38)(28,35,36,32)(34,40,39,37);(1,3,5,4)(2,6,7,8)(9,11,15,10)(12,19,22,16)(13,17,14,18)(20,23,21,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(26,31)(27,33)(32,35)(34,39);(9,13,15,14)(10,17,11,18)(12,20,22,21)(16,23,19,24)(25,28,29,36)(26,32,31,35)(27,34,33,39)(30,37,38,40);(1,4,5,3)(2,6,7,8)(9,13,15,14)(10,18,11,17)(12,21,22,20)(16,23,19,24);(9,14,15,13)(10,17,11,18)(12,21,22,20)(16,23,19,24)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548990);